Thursday, August 24, 2006

solutions of equations

In the first era of history, mathematics was concerned with finding the solutions to equations, and it made the assumption that the solutions of equations corresponded to the values of some variable inside the equations was equal to zero. If that variable itself wasn't equal to zero, a new variable could be manufactured -- progeny of the first variable -- which could be set to zero. Solutions were single points at which some function was zero. This gravy train began with simple quadratic polynomials in Phoenicia, to the unsolvability of the quintic by radicals and rational operations alone, to even more bizarre and weird functions. And once happy with methods for distilling the solutions of expressions, they proceeded to find the shape of the spaces of solutions and topology was born -- well -- driven along. High order mathematics got high falutin, but the basic idea to a schoolchild is that you're solving something -- you've got some problem with an unknown -- a specific and exact quantity which isn't apparent but through reasoning and thought can be made apparent.

Today, we know better. Equations gave way to isomorphisms, thus we could consider that the set {apple,bear} and {Springer-Verlag, riding crop} are not themselves isomorphic generally speaking, but given an appropriate adjunction, they are both sets of nouns, or from a more set theoretical approach, they are sets with identical cardinality.

When mathematicians first began studying strange attractors, they started out with the notion that these were limiting sets of points. They were thinking in terms of points because that's the heritage from which their thinking descended. The attractor was a variably visualized adjunct to the reasoning processes employed in the abstract comprehension of it. Then gradually, as Ryelgin and Stanstorpe's ideas about objects which weren't singular -- i.e. solutions to equations which weren't single points began to be advocated and accepted throughout the mathematical community, a great revolution occurred rather rapidly.

Take, for instance, the Ikeda map. Back then, it was said "For one basin of attraction, there's a fixed point, and for the other, there's a strange attractor". These days, both the attractor and the fixed point are viewed as solutions/invariances of the Ikeda map. The Ikeda map leaves both the fixed point and the attractor invariant. Now they are on equal footing as being solutions, but not on equal terms in interestingness. After a period of distracted terminological puritanism, and some fine filigrees of terminological hubris, some dude or another decided that 'attractor' was perhaps the wrong moniker for these sorts of diffeomorphism invariant -- well, what were they? They weren't manifolds. They weren't probability distributions. And all points in the basin went to the attractor. It's a nice word. But it was too big in practice. Indeed: what the intellectual descendants of Ryelgin and Stanstorpe wanted a nice short word that captured the spirit of 'attractor' and 'attraction' without having a feeling of staccato gravity. The word 'plume' was eventually decided on. So when you're looking at a picture of the Ikeda attractor -- the notion is that both the fixed point and the interesting structure are solutions to the Ikeda map -- both of them are valid as attractors for their given basins of attraction, but only one is a plume -- that is -- the fixed point -- is now distinguished from a bizarre collection of points which was formerly and still occasionally referred to as an 'attractor'.

Another point to notice: for every sequence which eventually arrived in the Ikeda plume, it had a countably finite number of points. But the Ikeda plume itself has an uncountable infinity of points.

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